Properties

Label 8036.15
Modulus $8036$
Conductor $8036$
Order $280$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, base_ring=CyclotomicField(280))
 
M = H._module
 
chi = DirichletCharacter(H, M([140,200,259]))
 
pari: [g,chi] = znchar(Mod(15,8036))
 

Basic properties

Modulus: \(8036\)
Conductor: \(8036\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(280\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8036.eo

\(\chi_{8036}(15,\cdot)\) \(\chi_{8036}(71,\cdot)\) \(\chi_{8036}(183,\cdot)\) \(\chi_{8036}(211,\cdot)\) \(\chi_{8036}(239,\cdot)\) \(\chi_{8036}(463,\cdot)\) \(\chi_{8036}(603,\cdot)\) \(\chi_{8036}(827,\cdot)\) \(\chi_{8036}(855,\cdot)\) \(\chi_{8036}(967,\cdot)\) \(\chi_{8036}(995,\cdot)\) \(\chi_{8036}(1051,\cdot)\) \(\chi_{8036}(1135,\cdot)\) \(\chi_{8036}(1163,\cdot)\) \(\chi_{8036}(1219,\cdot)\) \(\chi_{8036}(1247,\cdot)\) \(\chi_{8036}(1331,\cdot)\) \(\chi_{8036}(1359,\cdot)\) \(\chi_{8036}(1387,\cdot)\) \(\chi_{8036}(1611,\cdot)\) \(\chi_{8036}(1751,\cdot)\) \(\chi_{8036}(1975,\cdot)\) \(\chi_{8036}(2003,\cdot)\) \(\chi_{8036}(2031,\cdot)\) \(\chi_{8036}(2115,\cdot)\) \(\chi_{8036}(2143,\cdot)\) \(\chi_{8036}(2199,\cdot)\) \(\chi_{8036}(2227,\cdot)\) \(\chi_{8036}(2283,\cdot)\) \(\chi_{8036}(2311,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{280})$
Fixed field: Number field defined by a degree 280 polynomial (not computed)

Values on generators

\((4019,493,785)\) → \((-1,e\left(\frac{5}{7}\right),e\left(\frac{37}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 8036 }(15, a) \) \(1\)\(1\)\(e\left(\frac{5}{56}\right)\)\(e\left(\frac{9}{140}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{237}{280}\right)\)\(e\left(\frac{69}{280}\right)\)\(e\left(\frac{43}{280}\right)\)\(e\left(\frac{107}{280}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{9}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8036 }(15,a) \;\) at \(\;a = \) e.g. 2